ESSC v33: Relational Coordinate Mapping for Structural Readout — Profile-Area Geometry, Equal-Complexity Controls, and Nonlinear Recoverability in 171 SPARC Galaxies

This study examines whether relational coordinates derived from stellar and gas radial-profile geometry can provide a compact and interpretable structural representation of galaxy rotation-profile data. The principal relational vector, REL3, consists of three predeclared profile-area coordinates: 1. global profile nestedness,2. half-area-radius difference,3. profile-center distance. The analyses were conducted using a processed SPARC-based dataset containing 171 galaxies and 3,375 radial data points. Validation was performed using whole-galaxy held-out cross-validation with 5 folds and 10 repeats. Performance was evaluated under both point-uniform and radial-support-weighted regimes, with equal total training weight assigned to each galaxy. The public record includes: - the final manuscript PDF,- the LaTeX manuscript source and figures,- three frozen analysis scripts,- derived galaxy-level profile-area geometry summaries,- profile-area geometry audit results,- equal-complexity control results,- nonlinear recoverability results,- statistical summaries,- provenance and audit manifests,- a public file manifest describing the contents of the package. ## Principal findings 1. REL3 is exactly recoverable from its primitive profile-area inputs to numerical precision. 2. REL3 therefore does not constitute an informationally independent physical variable. 3. REL3 improves Ridge-model prediction relative to the base model under radial-support-weighted evaluation. 4. The REL3 increment is not consistently superior to equal-count separate-area or nonrelational control feature sets. 5. REL3 does not retain a confirmed predictive increment beyond its primitive inputs in the nonlinear-model audit. 6. REL3 is therefore interpreted as an efficient and interpretable nonlinear structural coordinate rather than as a new source of physical information. 7. Stronger results observed in some bulge-present and densely sampled galaxy subsets are treated as conditional findings requiring independent confirmation. The statistical analyses use galaxy-level paired bootstrap and sign-flip inference with a global Benjamini-Hochberg false-discovery-rate correction. This global correction is intentionally conservative. Train-only feature ranking for selected control models used an internal split within each outer training fold. The identities of individually selected control features are therefore procedure-dependent and are not interpreted as stable physical discoveries. Near-zero cumulative log-ratio coordinates were operationally regularized for numerical stability and were not assigned direct physical meaning. The fixed REL3 vector does not contain cumulative log-ratio features. ## Claim boundaries This study does not claim: - a new law of galaxy dynamics,- a replacement for dark matter,- a new theory of gravity,- a causal physical coupling operator,- informational independence of REL3,- measurement of actual two-dimensional galactic disk area,- temporal reconstruction of galaxy evolution,- pristine external validation. The profile-area quantities used here are one-dimensional descriptive proxies derived from processed radial tracer profiles. Predictive improvement does not by itself establish causality, physical coupling, or a fundamental physical law. ## Data availability and redistribution boundary The analyses use processed galaxy rotation-profile data derived from the SPARC dataset. The processed point-level input file, `POINT_NATIVE_AND_COMMON_SUPPORT_FEATURES.csv`, is not redistributed in this record because an explicit redistribution license for the underlying point-level SPARC-derived content was not confirmed. Users wishing to perform a complete rerun must obtain the relevant source data from the official SPARC source, cite the SPARC master paper and the original rotation-curve sources where appropriate, and reconstruct the required processed point-level input table. The public package includes derived galaxy-level summaries and frozen audit outputs sufficient for inspection of the reported results, but it is not a standalone raw-data reproduction package. ## Re-execution status The included scripts are provided as frozen analytical records. They are not directly executable from the public package alone because the processed point-level input table is intentionally not redistributed. The package documents the required filenames, workflow dependencies, feature definitions, validation design, statistical procedures, output structure, and claim boundaries. ## AI-assisted preparation OpenAI’s ChatGPT was used as an assistive tool for English-language drafting, manuscript organization, code development and review, and presentation of statistical audit results. The research questions, ESSC framework, analytical decisions, execution of the analyses, interpretation of results, and final conclusions were determined and approved by the author. ChatGPT is not an author and was not used as a source of scientific evidence.